Actions of dense subgroups of compact groups and $\textrm{II}_1$-factors with the Haagerup property

Abstract

Let $M$ be a finite von Neumann algebra with the Haagerup property, and let $G$ be a compact group that acts continuously on $M$ and that preserves some finite trace $\tau$. We prove that if $\Gamma$ is a countable subgroup of $G$ which has the Haagerup property, then the crossed product algebra $M\rtimes\Gamma$ has also the Haagerup property. In particular, we study some ergodic, non-weakly mixing actions of groups with the Haagerup property on finite, injective von Neumann algebras, and we prove that the associated crossed products von Neumann algebras are $\textrm{II}_1$-factors with the Haagerup property. If moreover the actions have Property $(\tau)$, then the latter factors are full.